Notes: Monday 22 March 2004

Schedule

• Since we will not cover everything you need to know for PS6 in class until Monday, you now have two options for Problem Set 6:
  • Turn it in on Monday, 22 March as originally scheduled, but without answering questions 7 and 8.
  • Turn it in on Monday, 29 March (one week later, answering all questions).
• Monday, 22 March: Read GEB, Aria with Diverse Variations and Chapter 13. This chapter proves that the halting problem is not decidable, and introduces the Church-Turing Thesis (which we will explore more in later classes). You will not be assigned to read Chapter XIV, but it goes into more depth on Gödel's proof and is recommended.
• Problem Set 7 will be out on 29 March and due on 7 April.
• Problem Set 8 will be out on 7 April; part I will be due on 12 April, part II will be due on 26 April (last day of class).
• Exam 2 will be handed out on 14 April and due on 19 April.

Notes
Procedure: A precise description of a process.

Alrogithm: A procedure that always terminates.

Proof: A proof of S in an axiomatic system is a sequence of strings, T₀, T₁, ..., T_n where:

• The first string is the axioms
• For all i from 1 to n, T_n is the result of applying one of the inference rules to T_n-1
• T_n is S

Proof Checking Problem

Input: an axiomatic system (a set of axioms and inference rules), a statement S, and a proof P containing n steps of S.

Output: true if P is a valid proof of S; false otherwise.

How much work is a proof checking procedure?

Finite-Length Proof Finding Problem

Input: an axiomatic system (a set of axioms and inference rules), a statement S, n (the maximum number of proof steps)

Output: A valid proof of S with no more then n steps if there is one. If there is no proof of S with <= n steps, unprovable.

How much work is a straightforward proof finding procedure?

n — maximum number of steps
r — number of inference rules

At worst, we can try all possible rules at every step:

Proof Finding Problem

Input: an axiomatic system, a statement S.

Output: If S is true, output is a valid proof. If S is not true, output is false.

Can we write an algorithm to solve the proof finding problem?

Computability

Is there an algorithm (a procedure that always terminates) that solves the problem?

A problem is decidable is there exists an algorithm that can solve the problem for all possible inputs. It is not necessary to know what that algorithm is to say a problem is decidable, only to know that some algorithm to solve it must exist. For example, chess is a decidable problem, even if we do not yet know an algorithm that solves it.

A problem is undecidable is there is no algorithm that can solve the problem. There might be a procedure, but it is not guaranteed to terminate.

Is the proof finding problem decidable?

How can you prove a problem is undecidable?

Halting Problem

Input: a procedure P (described by a Scheme program), and the input to that procedure
Output: true if applying P to input halts (finishes execution), false otherwise.

(define (halts? procedure input) ...?...)

What if we had halts?

(define (find-proof S axioms rules)
  ;; If S is provable, evaluates to a proof of S.
  ;; Otherwise, evaluates to #f.
  (if (halts? find-proof-exhaustive 
	      S axioms rules)
      (find-proof-exhaustive S axioms rules)
      #f))

Where (find-proof-exhaustive S axioms rules) is a procedure that tries all possible proofs starting from the axioms that evaluates to a proof if it finds one, and keeps working if it doesn't.

We can prove the halting problem is undecidable informally by arguing that if we could define halts?, we could use it to define contradict-halts (the input parameter to contradict-halts is not used, but necessary because of how we defined halts?):

(define (contradict-halts input)
  (if (halts? contradict-halts null) 
      (infinite-loop)
      200))

But contradict-halts is non-sensical: if it halts, it loops infinitely; if it doesn't halt, it evaluates to 200 and halts. This means something in the program must not exist. I'm pretty sure 200 exists, and we know how to define infinite-loop, and if seems likely to exist (and its worked well for us so far). So, it must be halts? that cannot exist.

Undecidable Problems If solving a problem P would allow us to solve the halting problem, then P is undecidable — there is no solution to P, since we have proved there is no solution to the halting problem!

Links

• A Person Paper on Purity in Language by William Satire (a.k.a. Douglas R. Hofstadter)
• Alan Turing
• On Computable Numbers, with an Application to the Entscheidungsproblem — Turing's original (1936) paper that introduced the Halting Problem (tougher to read than Gödel's)
• Breaking the Code — movie about Alan Turing
• Halting Problem (Hockey Team)
• Time 100: Alan Turing

There is a remarkably close parallel between the problems of the physicist and those of the cryptographer. The system on which a message is enciphered corresponds to the laws of the universe, the intercepted messages to the evidence available, the keys for a day or a message to important constants which have yet to be determined. The correspondence is very close, but the subject matter of cryptography is very easily dealt with by discrete machinery, physics not so easily.
Alan Turing